Radial and Axial Components of Magnetic Force. 193 



Xow &&fr - nyda [ - 47r.3 + V x - V,]* ............... (1 ), 



where /* = the number of turns per unit length -per unit increase in 



radius, 



y = current in C.G.S. units, 



V-, = potential at the point considered due to a plane area of 

 surface density unity, coinciding with the positive end of 

 the coil, 

 V., = potential due to the negative end. 



If r and / be the distance of the point considered from the centres 

 of the positive and negative ends of the coil respectively, and P n and 

 Q n the nth zonal harmonics corresponding to the angles and < which 

 r and r' make respectively with the axis of the coil, we have 



when r < a, and 



= ' ^ _ t 



-'#* S-i '' 



when r > a, with similar expressions for V. 2 in terms of / and Q. 

 The following relations are true : 



m ^ P - 1 ^ P 



- ~ 



rP - I*- 1 -~ 



(fe a do 



(4) ;r-P- = -ai*-iP,_i.} 



When Q and r' are substituted for P and r, the forms of (1) and (2) 

 remain unaltered ; whilst (3) and (4) become respectively 



(6) j/-^ = <n'-'Q,_,. 



* Maxwell's f Electricity and Magnetism,' Second Edition, vol. 2, 676, eq. 12. 



f Ibid., eqs. 14 and 15. 



The I formulae (3) and (4) may be generalised as follows : 



(CT-/)! 



